Textbook Question
Use the appropriate reciprocal identity to find each function value. Rationalize denominators when applicable. See Example 1. csc θ , given that sin θ = ―3/7
Ch. 1 - Trigonometric Functions
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 2, Problem 12
Find the measure of each marked angle. In Exercises 19–22, m and n are parallel. See Examples 1 and 2 .
Verified step by step guidance1
Identify the given information: lines \(m\) and \(n\) are parallel, and there are marked angles formed by a transversal intersecting these parallel lines.
Recall the properties of angles formed by a transversal with parallel lines, such as corresponding angles, alternate interior angles, alternate exterior angles, and consecutive interior angles, which are either equal or supplementary.
Use the fact that corresponding angles are equal when two lines are parallel, so if you know one angle, you can find its corresponding angle on the other parallel line.
Apply the property that alternate interior angles are equal, which helps find unknown angles inside the parallel lines but on opposite sides of the transversal.
If necessary, use the fact that angles on a straight line sum to \(180^\circ\) to find missing angles by setting up equations and solving for the unknown angle measures.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Parallel Lines and Transversals
When two parallel lines are cut by a transversal, several angle relationships are formed, such as corresponding angles, alternate interior angles, and consecutive interior angles. These relationships help determine unknown angle measures by establishing equality or supplementary conditions.
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Example 1
Angle Relationships
Key angle pairs include corresponding angles (equal), alternate interior angles (equal), and consecutive interior angles (supplementary). Understanding these relationships allows you to set up equations to find unknown angles when parallel lines are involved.
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Using Algebra to Solve for Angles
Often, marked angles are expressed in algebraic terms. By applying angle relationships from parallel lines and transversals, you can form equations and solve for the variable, then substitute back to find the exact angle measures.
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Related Practice
Textbook Question
Find the measure of each marked angle.
Textbook Question
Find the measure of each marked angle.
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Textbook Question
Find the measure of (a) the complement and (b) the supplement of an angle with the given measure. See Examples 1 and 3. 30°
Textbook Question
Find the measure of each marked angle. In Exercises 19–22, m and n are parallel. See Examples 1 and 2 .
Textbook Question
Sketch an angle θ in standard position such that θ has the least positive measure, and the given point is on the terminal side of θ. Then find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. See Examples 1, 2, and 4. (―12 , ―5)
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